We will continue with the treadmill data set introduced in Chapter 1 and the SLR fit in the practice problems in Chapter ??. The following code will get you back to where we stopped at the end of Chapter ??:
treadmill <- read_csv("http://www.math.montana.edu/courses/s217/documents/treadmill.csv")
plot(TreadMillOx~RunTime, data=treadmill)
tm <- lm(TreadMillOx~RunTime, data=treadmill)
summary(tm1)7.1. Use the output to test for a linear relationship between treadmill oxygen and run time, writing out all 6+ steps of the hypothesis test. Make sure to address scope of inference and interpret the p-value.
7.2. Form and interpret a 95% confidence interval for the slope coefficient “by hand” using the provided multiplier:
## [1] 2.045237.3. Use the confint function to find a similar confidence interval,
checking your previous calculation.
7.4. Use the predict function to find fitted values, 95% confidence, and 95%
prediction intervals for run times of 11 and 16 minutes.
7.5. Interpret the CI and PI for the 11 minute run time.
7.6. Compare the width of either set of CIs and PIs – why are they different? For the two different predictions, why are the intervals wider for 16 minutes than for 11 minutes?
7.7. The Residuals vs Fitted plot considered in Chapter ?? should have suggested slight non-constant variance and maybe a little missed nonlinearity. Perform a log-transformation of the treadmill oxygen response variable and re-fit the SLR model. Remake the diagnostic plots and discuss whether the transformation changed any of them.
7.8 Summarize the \(\log(y) \sim x\) model and interpret the slope coefficient on the transformed and original scales, regardless of the answer to the previous question.